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Full text of "Men Of Mathematics"


Hermite's unexpected victory over the obstinate e inspired
mathematicians to hope that <* would presently be subdued in a
similar manner. For himself, however, Hermite had had enough
of a good thing. fcl shall risk no thing' , he 'wrote to Borchardt "on
an attempt to prove the transcendence of the number r. If
others undertake this enterprise, no one will be happier than I
at their success, but believe me, my dear friend, it will not fafl
to cost them some efforts.' Nine years later (in 1882) Ferdinand
Lindemann of the University of Munich, using methods very
similar to those which had sufficed Hermite to dispose of e,
proved that IT is transcendental, thus settling for ever the pro-
blem of 'squaring the circle'. From what Lindemann proved it
follows that it is impossible with straight-edge and compass
alone to construct a square whose area is equal to that of any
given circle  a problem which had tormented generations of
mathematicians since before the time of Euclid.

As cranks are still tormented by the problem, it may be in
order to state concisely how Lindemann's proof settles the
matter. He proved that ^ is not an algebraic number. But any
geometrical problem that is solvable by the aid of straight-edge
and compass alone, when restated in its equivalent algebraic
form, leads to one or more algebraic equations with rational
integer coefficients which can be solved by successive extrac-
tions of square roots. As -n- satisfies no such equation, the circle
cannot be 'squared' with the implements named. If other
mechanical apparatus is permitted, it is easy to square the
circle. To all but mild lunatics the problem has been completely
^dead for over half a century. Nor is there any merit at the
present time in computing w to a large number of decimal
places - more accuracy in this respect is already available than
is ever likely to be of use to the human race if it survives for a
billion to the billionth power years. Instead of trying to do the
impossible, mystics may like to contemplate the following
useful relation between e, v9  1 and V  1 till it becomes
as plain to them as Buddha's navel is to a blind Hindu